Automata and Differentiable Words
Abstract
We exhibit the construction of a deterministic automaton that, given k > 0, recognizes the (regular) language of k-differentiable words. Our approach follows a scheme of Crochemore et al. based on minimal forbidden words. We extend this construction to the case of C∈finity-words, i.e., words differentiable arbitrary many times. We thus obtain an infinite automaton for representing the set of C∈finity-words. We derive a classification of C∈finity-words induced by the structure of the automaton. Then, we introduce a new framework for dealing with ∈finity-words, based on a three letter alphabet. This allows us to define a compacted version of the automaton, that we use to prove that every C∈finity-word admits a repetition in C∈finity whose length is polynomially bounded.